### Estimation of Population Growth with Mobility Rate and Expansion Rate by Cubic Spline

2018 International Journal for Research in Applied Science and Engineering Technology
This paper explores the method of estimating the population growth in various time intervals at different locations. The nature of growth of population over the various locations is constructed by both mobility rate of population-movement and expansion rate of migrated population. Where, mobility rate of population-movement controls the over expansion of population, mainly at initial locations by expansion rate of migrated population such that the distribution of population at all locations
more » ... ins uniformly. For that, the Mathematical Model of Partial Differential Equation (PDE) is formed and its solution is derived by the numerical method; Cubic Spline Explicit and Implicit with different initial and boundary conditions, as a population growth prediction with respect to time and location. 1098 The equation (1) gives growth of population (P) in time t at place x. It will be determined by any analytical as well as numerical methods along with required initial and boundary conditions. Here we are following Cubic Spline Method for the solution of one dimensional partial differential equation. Also we are taking two different types of initial and boundary conditions for the prediction the population growth. Case-(i) P(x, 0) = p + p e , P(0, t) = p + p Case-(ii) P(x, 0) = p + p e , P(0, t) = p + p e III. SPLINE METHOD [6],[7] An equation which satisfies the own characteristic with any initial and/or boundary condition is called analytic solution. It is not necessary that every PDE has analytical solution therefor another approach is also required for solving PDE. There are numerous types of numerical approach exists to solve (partial differential equation) PDE. Here we will use cubic spline numerical method for the solution of one dimensional PDE. The method spline is nothing but it is piecewise connecting polynomials of any degree. There are many types of spline polynomials exists viz., linear spline, quadratic spline, cubic spline and quantic spline etc. Out of these, the cubic spline of degree three has been found the most popular approximation method. Hence, we are adopting cubic spline method and replacing all the PDE term by approximation terms. Also the method has two parts for calculating the PDE, one is Explicit Method of Cubic Spline and another is Implicit Method of Cubic Spline. In Explicit Method we use forward difference and central difference formulas for the LHS of eq. (1) and replacing RHS term of equation (1) by the second derivatives S , . In Implicit Method same formula will be followed which we used in Explicit Method but here the second order partial derivative term of equation is replacing by average of S , , S , . A. Explicit Method [6], [7]